A Few Facts About Quasiconformal Mappings
If then the maps and are both quasiconformal and have constant dilatation .
If then the map is quasiconformal (here is a complex number) and has constant dilatation . When, this is an example of a quasiconformal homeomorphism that is not smooth. If, this is simply the identity map.
A homeomophism is 1-quasiconformal if and only if it is conformal. Hence the identity map is always 1-quasiconformal. If is K-quasiconformal and is K' -quasiconformal, then is K K' -quasiconformal. The inverse of a K-quasiconformal homeomorphism is K-quasiconformal. Hence the set of quasiconformal maps forms a group under composition.
The space of K-quasiconformal mappings from the complex plane to itself mapping three distinct points to three given points is compact.
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